350 
PEOFES>SOR A. R. I’ORSYTH ON THE DIFFERENTIAL INVARIANTS 
to modify the form of the equations (IIj'. . . (IIq/ by removing from them all the 
terms that involve those derivatives. The substituted derivatives with respect to 
a, h, c, a', h', c' do not occur~a result only to be expected, because these quantities 
are simultaneous solutions of all the six equations. Consequentlv, in any differential 
operations, the quantities a, h, c, a’, c' behave like constants. 
In order that 
/ zz: rq -j- -p ])Q^ _j_ ^ 
where Pj, Q^, Pq, are functions of and of the first derivatives of E, F, G, 
but are indei)endent of the quantities w and v, 6, a, h, c, h', c', may satisfy 
we must have 
- S A.= -3. 
2E 4- F 
0Fqo ^ c)F,o- 
2E L+F,L)Q, = 0, 
2E + A. 0. 
2eA +f^|;)S_ = E„,. + 2E„9. 
0E, 
no 10 ' 
In order that the same quantity may satisfy we must have 
' + Cl P^ = 0 , 
SEio oFio/ 
2F + G 
0E ^ 
-< 0 
10 
0F 
10 ' 
V 
-i 0 
2F - + G 
Qi — 2 
Ri = 0, 
' - 3F + V - 2F.„) s. 
Similarly, tlie equation (lE)^ re(pures 
. 0 
EA +2F V)« = 0, 
0F 
01 
and the equation (ITc)' requires 
F + -G ^ ' ) 0 = 0 
or,„ c’Gni 
01 '-'-^01 
for © = Pj, Q^, Pj,j Sj. The equation (lE)^ requires 
^ +fA +OF 3 'u,= 
-'(II '-'-^10 ^^01 ^'^^10 
SF + 4 + 4 + -f' 44) =« 
cdio cdq„ 0 G,o/ 
lor <h = Pj, Qi, R^, and 
